For a binary classification task, it has been shown [1]}, [2]} that \(r\)  and \(\operatorname{\mathbb {E}}\left[A\right]\)  are linearly related averaged over all possible true-positive rates. They showed that:
\(\operatorname{\mathbb {E}}\left[A\right] = \frac{P}{|S^p|+|S^n|} (1-\frac{P}{|S^p|+|S^n|}) (2r-1) + \frac{1}{2},\) 
